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Unformatted text preview: Atsushi Inoue ECG 765: Mathematical Methods for Economics Fall 2009 Lecture Notes 2 Sequences and Limits To characterize objective functions and constraint sets in optimization problems, we need to introduce some mathematical tools. Outline A. Functions B. Sequences and Limits: The Scalar Case C. Sequences and Limits: The Vector Case A. Functions Definition. Let X and Y be two sets, and suppose that with each element x of X there is associated, in some manner, an element of B , which we denote by f ( x ). Then f is said to be a function from X to Y (or mapping of X into Y ). This is often denoted by f : X Y . The set A is called the domain of f (we also say f is defined on X ), and the elements f ( x ) are called the values of f . The set of all values of f is called the range of f . B. Sequences and Limits: The Scalar Case Definition. Let { x 1 ,x 2 ,...,x n ,... } be a sequence of real numbers and let x be a real number. We say that x is the limit of this sequence if for any small positive number , there is a positive integer N such that for all n N , x n is in the -interval about x ; that is, | x n- x | &lt; ....
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